# Coherent Potential Approximation as a Voltage Probe

**ABSTRACT** Coherent potential approximation (CPA) has widely been used for studying

residual resistivity of bulk alloys and electrical conductivity in

inhomogeneous systems with structural disorder. Here we revisit the single-site

CPA within the Landauer-B\"uttiker approach applied to the electronic transport

in layered structures and show that this method can be interpreted in terms of

the B\"uttiker's voltage-probe model that has been developed for treating phase

breaking scattering in mesoscopic systems. We demonstrate that the on-site

vertex function which appears within the single-site CPA formalism plays a role

of the local chemical potential within the voltage-probe approach. This

interpretation allows the determination of the chemical potential profile

across a disordered conductor which is useful for analyzing results of

transport calculations within the CPA. We illustrate this method by providing

several examples. In particular, for layered systems with translational

periodicity in the plane of the layers we introduce the local resistivity and

calculate the interface resistance between disordered layers.

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**ABSTRACT:**We report the investigation of conductance fluctuation and shot noise in disordered graphene systems with two kinds of disorder, Anderson type impurities and random dopants. To avoid the brute-force calculation which is time consuming and impractical at low doping concentration, we develop an expansion method based on the coherent potential approximation (CPA) to calculate the average of four Green's functions and the results are obtained by truncating the expansion up to 6th order in terms of "single-site-T-matrix". Since our expansion is with respect to "single-site-T-matrix" instead of disorder strength $W$, good result can be obtained at 6th order for finite $W$. We benchmark our results against brute-force method on disordered graphene systems as well as the two dimensional square lattice model systems for both Anderson disorder and the random doping. The results show that in the regime where the disorder strength $W$ is small or the doping concentration is low, our results agree well with the results obtained from the brute-force method. Specifically, for the graphene system with Anderson impurities, our results for conductance fluctuation show good agreement for $W$ up to $0.4t$, where $t$ is the hopping energy. While for average shot noise, the results are good for $W$ up to $0.2t$. When the graphene system is doped with low concentration 1%, the conductance fluctuation and shot noise agrees with brute-force results for large $W$ which is comparable to the hopping energy $t$. At large doping concentration 10%, good agreement can be reached for conductance fluctuation and shot noise for $W$ up to $0.4t$. We have also tested our formalism on square lattice with similar results. Our formalism can be easily combined with linear muffin-tin orbital first-principles transport calculations for light doping nano-scaled systems, making prediction on variability of nano-devices.Journal of Applied Physics 05/2013; 114(6). · 2.21 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Accounting for the effects of disorder on the transport properties of electronic devices is indispensable for comparison with experiment. However, theoretical treatment of disorder presents essential difficulty because the disorder breaks the periodicity of the system. The coherent potential approximation (CPA) solves this problem by replacing the disordered medium with a periodic effective medium. However, calculating the electron current within CPA requires summing scattering diagrams to infinite order called vertex corrections. In this work we reformulate CPA for nonequilibrium electron transport. This approach, based on the nonequilibrium Green's function formalism, greatly simplifies the treatment of disordered transport by eliminating the vertex corrections.Physical Review B 01/2012; 85(23):235111. · 3.66 Impact Factor - SourceAvailable from: Kartsev Alexey[Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the static and dynamical behavior of 1D interacting fermions in disordered Hubbard chains, contacted to semi-infinite leads. The chains are described via the repulsive Anderson-Hubbard Hamiltonian, using static and time-dependent lattice density-functional theory. The dynamical behavior of our quantum transport system is performed via an integration scheme available in the literature, which we modify via the recursive Lanczos method, to increase its efficiency. To quantify the degree of localization due to disorder and interactions, we adapt the definition of the inverse participation ratio to obtain an indicator which is both suitable for quantum transport geometries and which can be obtained within density-functional theory. Lattice density functional theories are reviewed and, for contacted chains, we analyze the merits and limits of the coherent-potential approximation in describing the spectral properties, with interactions included via lattice density functional theory. Our approach appears to able to capture complex features due to the competition between disorder and interactions. Specifically, we find a dynamical enhancement of delocalization in presence of a finite bias, and an increase of the steady-state current induced by inter-particle interactions. This behavior is corroborated by results for the time-dependent densities and for the inverse participation ratio. Using short isolated chains with interaction and disorder, a brief comparative analysis between time-dependent density-functional theory and exact results is then given, followed by general conclusive remarks.Physical review. B, Condensed matter 04/2012; 87(11). · 3.77 Impact Factor

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Coherent Potential Approximation as a Voltage Probe

M. Ye. Zhuravlev,1,2 A. V. Vedyayev,3 K. D. Belashchenko1, and E. Y. Tsymbal1*

1 Department of Physics and Astronomy, Nebraska Center for Materials and Nanoscience,

University of Nebraska, Lincoln, Nebraska 68588, USA

2 Kurnakov Institute for General and Inorganic Chemistry,

Russian Academy of Sciences, 119991 Moscow, Russia

3 Department of Physics, M. V. Lomonosov Moscow State University, 119899 Moscow, Russia

Abstract

Coherent potential approximation (CPA) has widely been used for studying residual resistivity of

bulk alloys and electrical conductivity in inhomogeneous systems with structural disorder. Here

we revisit the single-site CPA within the Landauer-Büttiker approach applied to the electronic

transport in layered structures and show that this method can be interpreted in terms of the

Büttiker’s voltage-probe model that has been developed for treating phase breaking scattering in

mesoscopic systems. We demonstrate that the on-site vertex function which appears within the

single-site CPA formalism plays a role of the local chemical potential within the voltage-probe

approach. This interpretation allows the determination of the chemical potential profile across a

disordered conductor which is useful for analyzing results of transport calculations within the

CPA. We illustrate this method by providing several examples. In particular, for layered systems

with translational periodicity in the plane of the layers we introduce the local resistivity and

calculate the interface resistance between disordered layers.

PACS: 73.63.-b, 73.23.-b, 73.50.Bk, 73.40.-c

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1. Introduction

Coherent potential approximation (CPA)1 is a powerful method for studying materials where

substitutional disorder determines their electronic properties. The CPA results from the self-

consistent solution of the quantum-mechanical multiple-scattering problem, which allows

treating disorder in terms of the configuration-averaged scattering matrix. Typically such a

solution is obtained within the single-site approximation, in which the properties of all sites but

one in the system are averaged over, and that one is treated exactly.2 This approach has been

broadly used for the description of short-ranged scattering in binary alloys.3,4 The CPA goes

beyond the limits of low concentration and weak scattering in a physically realistic way,

providing self-consistency of the solution.

The CPA has also become a useful tool for studying transport properties of alloys and

disordered systems within the linear response theory.5 The configurational averaging in this case

requires averaging of the two one-electron Green’s functions GG , in contrast to equilibrium

properties that are simply determined by G . Thus, the extension to the linear response involves

the determination of the so-called “vertex corrections” GG

performed consistently with the single-site CPA resulting in a closed set of equations for the

conductance.5 Such an approach has been extensively used for calculating the residual resistivity

of binary alloys6,7,8,9 and layered structures,10,11 including the extension to treat realistically band

structures of disordered systems.12,13,14,15,16,17

The configurational averaging within the CPA substitutes an effective medium that possess

translational invariance for the original system which is characterized by a random non-periodic

potential. The CPA replaces the latter by the self-energy , which is an energy-dependent non-

Hermitian operator. Its real part shifts the energy levels of the undisturbed system, whereas the

imaginary part broadens the energy levels due to the finite scattering lifetime. Thus, the original

system which in the absence of inelastic scattering would describe phase-coherent propagation of

electronic waves is replaced by an artificial system that involves phase non-conserving

scattering.

This situation is analogous to that within the Büttiker’s voltage-probe model introduced to

treat dephasing in mesoscopic physics.18 This model involves fictitious voltage probes into an

otherwise coherent system, which produce phase-breaking processes. No net current flows in the

fictitious electrodes, and hence all electrons scattered into the voltage probes are emitted back

into the sample. Such a scattering process is incoherent, and phase memory of the scattered

electrons is completely lost. To realize this model in a practical calculation, fictitious probes are

attached to each site of the sample, and their chemical potentials are adjusted such that no net

current flows in the fictitious electrodes. Due to its appealing simplicity the voltage-probe model

has been extensively used for studying quantum transport in quantum dots,19,20 molecular

junctions,21,22 nanowires23,24 and other mesoscopic and nano systems.25,26,27,28 It has been shown

that there is an analogy between the voltage-probe method and imaginary-potential models for

dephasing.29

In this paper we revisit the CPA1,5 within the Landauer-Büttiker approach30,31 applied to the

electronic transport in layered structures and show that this method can be interpreted in terms of

the voltage-probe model.18 We demonstrate that the on-site vertex function which appears within

the single-site CPA formalism plays a role of a local chemical potential within the voltage-probe

approach. This interpretation allows the determination of a chemical potential profile across a

disordered sample which is useful for analyzing results of transport calculations within the CPA.

G G

. This calculation can be

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In particular, for layered systems with translational periodicity in the plane of the layers we

demonstrate the possibility of introducing the local resistivity and calculating the interface

resistance between disordered layers.

The paper is organized as follows. In section 2 we briefly outline the Büttiker’s voltage-

probe model. In section 3 we revisit the single-site CPA, and in section 4 derive expressions for

transmission within the Landauer-Büttiker approach. In section 5 we show that the CPA results

can be interpreted within the voltage-probe model. In section 6 we perform calculations for

particular layered systems. In section 7 we summarize the results.

2. Voltage-probe model

Following the Landauer-Büttiker approach 30,31 we consider a “sample” attached to two semi-

infinite electrodes. The electrodes are connected to reservoirs that are characterized by the

equilibrium Fermi distribution functions ( )

L

f E

the energy and

L

and

R

are the chemical potentials of the left and right electrodes

respectively. The electric current is driven in the sample by the applied voltage V, such that

eV

.

Within the Büttiker’s voltage-probe model,18 each atomic site of the sample is connected to

a fictitious electrode n that is characterized by the equilibrium Fermi function

with chemical potential

n

. The electrodes are allowed to float to different

current in the electrodes and thus local current conservation in the system.

Within the linear response the electric current at zero temperature in electrode p (p = L, R, or

n) is given by (see, e.g., ref. 32)

()

L

f E

and ( )E()

RR

ff E

, where E is

LR

( )()

nn

f E

, to ensure zero

f E

n

2

h

()

ppq pq

q

e

IT

, (1)

where summation is performed over all electrodes q (q = L, R, or n) and

between electrodes q and p. The latter can be calculated as follows

pq

T is the transmission

†

p

†

q

†

( ) (G)

pqpq

TTrG

,

(2)

where p and q are the self-energies associated with p and q electrodes respectively and G is

the (retarded) Green’s function33 of the sample coupled to the electrodes

1

( ) ( ) ( )

LRn

n

Here L(E), R(E), and n are the self-energies associated with the left, right and floating

electrodes, and summation is performed over floating electrodes (scattering sites of the sample).

The electric current (1) is obtained from transmission (2) calculated at the Fermi energy E = EF.

Chemical potentials of the floating electrodes are found by assuming that the electric current

flowing in each electrode is zero. Using Eq. (1) we find

G EEHEE

. (3)

0()()()

nLnL nRnR nmnm

m

TTT

. (4)

This equation may be interpreted as a local current conservation condition. The solution of this

system of linear equations determines chemical potentials at each site of the sample. Once the

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chemical potentials are found the net current passing through the sample is given by the current

in the left (or right) electrode:

2

h

()()

L LRLRLnLn

n

e

IITT

. (5)

Thus, the voltage-probe model introduces phase breaking scattering in the system ensuring

current conservation throughout the sample.

Below we apply this approach to layered structures which are infinite and translationally

periodic in the plane of the layers. In this case it is convenient to introduce the transverse wave

vector

||

,

xy

k k

k that is conserved during transmission across the sample. The Green’s

function of the system

||

(, )

GE

k

, as well as the self-energies of the left and right electrodes,

( , )

L

E

k

and

||

( , )

R

E

k

, become functions of

number, i.e.

1,...nN

, where N is the total number of layers, and assume that the self-energies

of the floating electrodes are layer dependent but constant within each layer. In this case the

summation over in-plane sites can be replaced by the respective integrations, so that the

transmission functions entering Eqs.(4) and (5) above are given by

|||| k . We use index n to characterize the layer

||

†

N

||1|| ||1||

2

||

†

n

||1||1||

2

||

†

Nn

||||||

2

||

†

nm

|| ||

2

()()()(),

(2 )

()()(),1,2,... ,

(2 )

()()(),1,2,... ,

(2 )

()(),, 1,

(2 )

I

L

I

RLRN

I

L

I

n Lnn

I

n

I

RnRnN

I

m

I

nmnmn

d

TGG

d

TGGnN

d

TGGnN

d

TGG m n

k

kkkk

k

kkk

k

kkk

k

kk 2,... .N

(6)

Here subscripts in the Green’s function G denote its matrix elements between different sites,

and we implicitly assume a single band model. In Eq. (6) we have defined

2Im

LL

, and

2Im

nn

. We have also assumed that the left and right electrodes are

coupled to the sample at sites 1 and N, respectively, so that the respective matrix elements of the

self-energy operators are

11LLnm

nm

and

3. Single-site CPA

2Im

I

RR

,

II

RR nNmN

nm

.

Now we outline the single-site CPA. We assume that there is a random potential U on each site

in the sample region. The CPA replaces the disordered system by an effective medium that is

described by complex coherent potential (self-energy)

n

n

, the components

n

being

dependent on layer

condition assumes that this Green’s function of the effective medium G is equal to the Green’s

function G averaged over disorder configurations, that is

1...nN

, but independent of a site within the layer. The self-consistency

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GG

, (7)

where ... denotes averaging over disorder configurations. This provides condition to find the

U as a perturbation, G can be written in terms of G

coherent potential Σ. Considering ()

()

GG G UG

. (8)

Averaging Eq. (8) and taking into account Eq. (7) we find

()0

UG

. (9)

Equivalently this equation can be expressed in terms of the T matrix34 which is defined by

GGGTG

, (10)

() ()

TUUGT

, (11)

and implies that

0

T . (12)

This equation can be solved within the single-site CPA, which introduces a single-site T-

matrix

iT according to

() ()

iiiiii

TUU GT

, (13)

where

scattering theory (see, e.g., ref. 35), T may be written as a sum of single-site contributions:

,

i

U and

iV are on-site random and coherent potentials. As follows from the multiple-

i

i

TQ

(14)

1

iij

j i

QTGQ

. (15)

These equations have a simple physical interpretation. The total scattered wave is a sum of

contributions from each atom given by the atomic T-matrix applied on an effective wave. The

effective wave consists of the incident wave and of the contribution to the scattered wave from

all other sites.

The single-site approximation assumes that the statistical correlation of

corresponding effective wave are negligible. Then we can decouple the averaging in Eq. (15) so

that Eqs. (14) and (15) average to

,

iT and of the

i

i

TQ

(16)

1

iij

j i

QTGQ

. (17)

Thus the self-consistency condition (12) becomes

0

iT

. (18)

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In our case, we have N non-equivalent sites within the sample, and hence Eq.(18) represents

a set of N coupled non-linear equations. For example, if we assume that disorder is formed by a

binary alloy characterized by on-site energy UA with a probability qA and on-site energy UB with

a probability qB (1

qq

), Eq. (18) reads

AB

0,1, 2,

1 (

) 1 (

)

AB

AB

nn

AB

n nnn nn

U

U

U

U

qqnN

GG

. (19)

Here

nn

G is the on-site Green’s function

||

||

2

( )

E

( , )(2 )

nn nn

d

GGE

k

k

, (20)

where

1

|||| ||

(, )E(, )E(, )E

LRn

n

GEH

kkk

. (21)

Eqs. (19) can be used to find

iterative procedure.

4. CPA transmission

n

. Normally these equations are solved numerically using an

Transmission across a disordered sample requires averaging over disorder. Using Eq. (2) for

transmission between left and right electrodes across the sample that is described by Green’s

function G we obtain

†I

L

I

R LR

TTTrGG

,

(22)

where we took in to account the fact that operator

entering Eq. (22) can be calculated within the single-site CPA using the approach

developed by Velický.5 Using Eqs. (10) for the Green’s function is terms of the T-matrix, and

taking into account that

R

does not depend on a random configuration and that according to Eq.

(12)

0T , we find

L

is configuration-independent. The average

†I

R

GG

†††I

R

I

R

GGGGG G

. (23)

Here operator known as the vertex correction is defined by

††I

R

TGG T

. (24)

Next, using Eqs.(16) and (17), can be represented as

,

nm

nm

(25)

where

††

n

I

R nmm

Q GG Q

, (26)

or

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††

s

††

11

I

Rnmmln

l m

s n

TGQGGQ GT

. (27)

Consistent with the single-site approximation of CPA we decouple Eq.(27) as follows

††

s

††

11

I

R nmmln

l m

s n

TGQGG Q GT

. (28)

Now we can take into account Eq. (18) saying that

0

iT

and the fact that variations of

n T on

different sites are statistically independent, so that

††

... ...

mnmnnn

TTTT

. In a similar way we

can conclude that

0

l Q

and

†

s

†

l

......

l lsl

QQQQ

, which leads to

†††

l

††I

R

I

Rmnmnnmnnln

l n

TGG G Q GG QGT

. (29)

Using the same approximation in Eq. (26) we obtain from Eq. (29)

†

mn

††

,

I

Rnnnlln

lm

l m l n

TGGGGT

. (30)

This is a system of equations which can be solved using an appropriate basis. For example,

within a single-band tight-binding model we define

nn

nn

, (31)

and

nn

Tn tn

. (32)

Thus, Eq. (30) is reduced to

††

nn

††

mn

†

ln

,

1()

I

R lmn n n

t t

nnn n

t t

nlnl l

l ml

G G GGGG

, (33)

where we included the diagonal term in the summation. We note that here indices l, m, and n

refer to sites and do not take into account periodicity of our system in the plane. Once the vertex

function is found the transmission can be obtained using Eqs.(22) and (23).

5. CPA as a voltage probe

Now, using the single-site CPA formalism we prove the following identity

2

††

Im1Im

nn n

t t

nnn n

t t

nn

GG

. (34)

According to Eq.(13) we have

†

2

11

()1

1

1

nnnnnnn

nnn

nnn

UTTT G

T G

T G

. (35)

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Taking an imaginary part and restructuring the terms we obtain

†

nn n

††

nn n

††

nn n

†

Im1 ImIm

nn nnn nnnn

T GG TT G G TTT G T

. (36)

Averaging over random configurations and taking into account

0

nT

, we find

2

††

nn n

†

Im1 Im

nn nnnn

T GTT G T

. (37)

Finally, using Eq. (32) we arrive at Eq. (34).

The identity (34) simplifies Eq. (33) which can now be written as follows

†

mn

†

ln

,

Im

Im

()

I

R lm

nn

nnl nll

l ml

n

G

GGGG

. (38)

Now, we exploit explicitly the periodicity of our system in the plane of the layers, using a mixed

representation

, n

k

, where

into account that

n

,

n

and

()()

RRnN mN

nm

kk

, we rewrite Eq. (38) as follows

|| k is the transverse wave vector and n is the layer number. Taking

nT are independent of a site in the plane and assuming that

|| ||

|| ||

†

Nn

†

mn

|||||||| ||

22

Im

Im

()()()()()

(2 )

(2 )

I

R

nn

n nNm nm

m

n

dd

G

GGGG

kk

kkkkk

. (39)

Here

The transmission is given by

nn

G is the on-site matrix element of the Green’s function within layer n given by Eq. (20).

|| ||

†

N

||1||||1 ||||1|| 1 ||

22

()()()()()()()

(2 )

(2 )

I

R

I

R

I

RNmmm

m

dd

TGGGG

kk

kkkkkkk

. (40)

Using definitions (6), expressions (39) and (40) can be rewritten as follows:

,

,

2 Im

I

nnnn nRm nm

T

m

GT

(41)

LRm Lm

T

m

TT

(42)

where

I

nnn

. (43)

Finally, we rewrite Eqs. (41) for the vertex functions

functions (6). For this purpose, we introduce the self-energy Σ whose matrix elements are

n entirely in terms of transmission

|||| ||11||

()()()()

mnnmn nmn LmnRmNnN

kkkk

. (44)

It is easy to see that

†

(2Im )2Im

I

n

I

nnR nLnmnn

nn

m

TTTGGG

, (45)

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where the integration over

|| k is implicitly assumed in the operator product in the square

brackets, and the latter equality follows from the identity:

we can rewrite Eqs.(41) as follows:

†

ImImGGG

. Using Eq. (45),

0( 1)(),1,2,...

n nLn nRnmnm

m

TTTnN

. (46)

These equations are identical to those given by formula (4) within the Büttiker’s voltage-

probe model.

measured with respect to the chemical potential of the left electrode set equal to zero, the

chemical potential of the right electrode being set equal to unity. This simply implies that

the reduced chemical potential given by

nn

then the transmission to the left electrode from the right electrode and all the floating electrodes.

The local current conservation condition requires that the local currents in all the floating

electrodes are zero. This is exactly what Eqs. (46) infer. Thus, the CPA vertex constants for

electric conductivity may be interpreted as local chemical potentials that provide zero current in

the floating electrodes.

6. Examples and discussion

n can be associated with a relative chemical potential of a floating electrode n

n is

/

L

eV

. The physical meaning of Eq. (42) is

While within the voltage-probe model the self-energies of the floating electrodes are

phenomenological parameters, the single-site CPA provides a clear recipe to determine the self-

energies. Once the type of disorder is known the self-energies can be found according to a self-

consistent procedure that provides on average a zero on-site T-matrix. In that sense, the proven

equivalence between the CPA and the voltage-probe model may be considered as a concrete

physical example where the voltage-probe model is justified.

The variation of the local chemical potential across the sample implies the presence of the

internal electric field. It is known that such a field may be used instead of the vertex corrections

for conductivity to provide the local current conservation.36 For example, such an approach was

used to calculate the conductance and magnetoresistance of segmented nanowires in the presence

of diffuse scattering.37 We have shown within the CPA how to calculate the local chemical

potential and thus the internal electric field.

The local chemical potential allows obtaining a useful insight into the transport behavior

within the CPA. The variation in the chemical potential across the conductor may be used to

determine the local resistivity and thus identify regions in an inhomogeneous sample

contributing differently to the resistance. This approach also allows finding the interface

resistance between the regions separating two disordered conductors. Below we consider a few

examples.

We calculate the conductance by considering a sample of disordered material which consists

of N layers and is connected to two perfect semi-infinite electrodes. We assume that the sample

is a binary alloy characterized by on-site energy UA with a probability qA and on-site energy UB

with a probability qB. The coherent potential of the system can be found by solving self-

consistently Eqs. (19). The Green’s function of the sample connected to the electrodes within the

CPA is given by Eq. (21). We assume a single-band tight-binding model and simple cubic lattice.

In this case, the eigenvalues of the Hamiltonian H are given by

where

0

t

is the hopping integral between neighboring sites and a is the lattice constant. The

||

() 2 (costcos)

xy

k ak a

k

,

Page 10

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self-energies of the electrodes are expressed through the surface Green’s functions of the leads

16

(a)

and are given by38

2

2

, ||||

( ,)()4 / 4

L REEEt

kk

.

4

8

12

0 1020 304050

0.00

0.05

0.10

0.15

0.20

0.25

-0.30

1.0

-0.25

-0.20

-0.15

0.0

1.5

0.2

0.4

0.6

0.8

Without Vertex Correction

With Vertex Correction

Resistance (a

2/G0)

(b)

Without Vertex Correction

Conductance (G0/a

2)

Layer thickness (a)

Vertex Correction

contribution

(c)

Coherent Potential (t)

Re()

Im()

(d)

Chemical Potential

0 10

Atomic layer number

2030 4050

0.0

0.5

1.0

(e)

Resistivity (a/G0)

Fig. 1: Results of transport calculations for a binary alloy layer placed between two semi-infinite

electrodes. (a) Areal resistance as a function of layer thickness with and without vertex

corrections; (b) Areal conductance as a function of layer thickness showing separately the

conductance without vertex corrections and the vertex correction contribution; (c) Real and

imaginary parts of the coherent potential across the sample; (d) Reduced chemical potential and

(e) local resistivity across the sample. Parameters used in the calculations: EF = 2t, UA = 0.6t,

UB = 1.4t, qA = 0.7, qB = 0.3. In figures (c), (d), and (e) the layer thickness is 50a.

the conductance quantum, and a is the lattice constant.

Fig.1 shows results of the calculation for a binary alloy layer embedded between two semi-

infinite electrodes. Here we set E = EF = 2t, UA = 0.6t, UB = 1.4t, qA = 0.7, qB = 0.3 which

provides a relatively weak disorder in the alloy. It is seen from Figs. 1a and 1b (red curves) that

ignoring the vertex correction in the transport calculation leads to an exponential increase of the

2

0

2/Geh

is

Page 11

11

areal resistance R (Fig. 1a) and decrease in the conductance G per unit area (Fig. 1b) with

disordered layer thickness. It has been shown that the CPA conductance without vertex

corrections is similar to the ballistic contribution which conserves

transmission across a disordered region.39 This contribution decreases exponentially on a scale

determined by the mean free path as determined by imaginary part of the CPA self-energy (the

blue curve in Fig. 1c). The vertex corrections restore the Ohm’s law making the resistance to

increase linear with layer thickness (the blue curve in Fig. 1a). As is seen from Fig. 1b, the vertex

contribution first increases with layer thickness, reaches maximum, and then decreases inversely

proportional to layer thickness. This behavior reflects a diffusive contribution to the

conductance, where elastic scattering involves scattering events between different

the increase of the diffusive part on the scale of the mean free part. Further increasing of the

layer thickness enhances the diffusive contribution to the resistance proportional to the number

of scattering events (layer thickness).

Since the alloy is assumed to be homogeneous the only inhomogeneity in the system occurs

near the interfaces between the disordered layer and perfect electrodes. This is reflected in the

coherent potential which is nearly constant across the layer, small variations being seen only near

the interfaces (Fig. 1c). The homogeneity of the bulk alloy is mirrored in the chemical potential

variation which drops linearly across the disordered region (Fig. 1d). The only sizable deviation

from the linear behavior occurs at the interfaces with electrodes where steps in the chemical

potential reflect the interface resistance (layers 0-1 and 50-51 in Fig. 1d).

In general, the chemical potential profile across a disordered inhomogeneous conductor may

be used to evaluate the local resistivity of the conductor, which may be useful for analyzing the

transport behavior. Since our system is quasi one-dimensional and the current is conserved, we

can define the local resistivity as follows:

,

|| k in the process of

|| k resulting in

( )z

d

dz

R

(47)

where R is the areal resistance of the whole system and it is assumed that the reduced chemical

potential is the continuous function of position z across the conductor. In our case of a discrete

lattice we can define the local resistivity, e.g., as follows

shows, as an example, the result of calculation of the resistivity for the system discussed above.

We see that the resistivity is nearly constant through the disordered layer, but has sharp features

near interfaces reflecting the interface resistance. There are weak oscillations in the resistivity

near the interfaces reflecting the quantum interference caused by interface perturbation.

In another example we consider a diffusive bilayer conductor representing two disordered

alloy layers (25 unit cells each) placed between two semi-infinite electrodes. Here we assume

that EF = 2t and disorder in the left (L) layer is fixed so that

0.65. The on-site atomic energies in the right (R) disordered layer are also fixed,

and

R

Ut

, whereas the relative concentration of the two alloy components,

varied. The results for the reduced chemical potential are displayed in Fig. 2a for different values

of

R

q . It is seen that with increasing disorder the voltage drop in the right segment becomes

more pronounced reflecting the increasing resistivity of this layer. This is evident from Fig. 2b

11

2/

nnnn

R a

. Fig. 1e

3

A

L

Ut

,

B

L

Ut

,

0.35

U

A

L q

, and

3

q , is

B

L q

A

R

t

BA

R

q and

B

R

A

Page 12

12

showing the site-dependent resistivity across the conductor. The resistivity of the left layer

constant, whereas the resistivity of the right layer

1.0

(a)

Chemical Potential

L

is

R

increases with alloying.

0.0

0.2

0.4

0.6

0.8

0.00.4

q

0.8

0

2

4

0102030 4050

0

1

2

3

4

(b)

Atomic layer number

Resistivity (a/G0)

q

A

R = 0.05

q

A

R = 0.25

q

A

R = 0.45

q

A

R = 0.65

A

R

Ri (a

2/G0)

Fig. 2: Results of transport calculations for a disordered bilayer system. (a) Reduced chemical

potential across the bilayer conductor for different concentration

Local resistivity across the bilayer. The inset shows the interface resistance between two

disordered layers as a function of alloying in the right segment. Parameters used in the

calculations: EF = 2t, 3

L

Ut

,

L

Ut

,

2

0

2/Geh

is the conductance quantum, and a is the lattice constant.

These results allow us to evaluate the interface resistance between two disordered layers.

The interface resistance has previously been derived in terms of transmission probabilities

between two ballistic electrodes assuming completely diffuse scattering in the bulk of the

layers.40 In our approach the diffuse scattering in the two adjacent layers is provided by the CPA.

In order to calculate the interface resistance, we fix two points in the conductor lying at distance

Lt from the interface in the left layer and at distance

areal resistance of the sample between the two points can be written as follows:

A

R

q in the right segment. (b)

AB

0.35

A

L q

, 0.65

B

L q

, 3

A

R

Ut

, and

B

R

Ut

.

Rt from the interface in the right layer. The

L L

t

iR R

tRR

, (48)

Page 13

13

where R is the areal resistance of the whole system and

iR is the interface resistance. By fixing

Lt and

resistivities of the left (L) and right (R) layers are nearly the same as in the bulk of these

layers, we can calculate the interface resistance from Eq. (48) given the known value of

between the two points. The result is displayed in the inset of Fig. 2b which shows

function of the degree of alloying in the right layer. It is known that depending on reflection

coefficients diffuse scattering can assist or suppress conduction across interfaces.41 In our case

the interface resistance slightly decreases with disorder which is due to opening new

transmission channels across the interface.42 This approach to calculate the interface resistance in

the presence of disorder may be considered as an alternative to that based on the supercell

calculation43,44 and the Boltzmann equation.45

1.0

2/G0)

Rt , so that the two points lie sufficiently far away from the interface and hence the local

iR as a

0.0

0.2

0.4

0.6

0.8

0.0 0.2

q

0.4

0

20

40

010 20304050

1

10

100

(b)

Atomic layer number

Resistivity (a/G0)

q

A

M = 0.05

q

A

M = 0.25

q

A

M = 0.45

(a)

Chemical Potential

A

M

Ri (a

Fig. 3: Results of transport calculations for a disordered trilayer system. (a) Reduced chemical

potential across the trilayer conductor system for different concentration

segment. (b) Local resistivity across the trilayer. The inset shows the interface resistance

between the left (right) and middle disordered layers as a function of alloying in the middle

layer. The parameters used in the calculation are as follows:

L R

q

,

4

M

Ut , and

2

M

Ut.

0

2/Geh

constant.

A

M

q in the middle

,

3

A

L R

Ut ,

,

L R

B

Ut,

,

0.6

A

L R

q

,

,

0.6

B

AB

2

is the conductance quantum, and a is the lattice

Page 14

14

Finally, we consider a disordered trilayer system where the left (L) and right (R) conducting

layers of thickness 16a are separated by a conducting middle (M) layer of thickness 18a. The

parameters characterizing the left and right layers are assumed to be identical, i.e.

L R

q

, and

,

0.6

L R

q

. We study transport properties of the system as a function of

alloying

M

q in the middle layer for which the on-site atomic energies of the alloy components

are assumed to be 4

M

Ut and 2

M

Ut. Fig. 3a shows the resulting variation of the reduced

chemical potential across the trilayer. With increasing

layer is increasing reflecting the increasing resistivity of this layer. The latter fact is also evident

from the site-dependent resistivity plots shown in Fig. 3b. Steps are seen at the interfaces

between the middle and adjacent layers as the result of the interface resistance. By performing a

calculation similar to that for the bilayer system we find that in this case the interface resistance

increases significantly with concentration

M

q in the middle layer alloy (see the inset in Fig. 3a).

This is due to a large mismatch between the on-site energy

energies in the left (right) layer alloys which leads to the large potential step at the interface with

increasing

M

q .

7. Summary

,

3

A

L R

Ut ,

,

L R

B

Ut,

,

0.4

AB

A

AB

A

M

q the voltage drop across the middle

A

A

M

U and the respective on-site

A

This work links the coherent potential approximation that has been widely used to describe the

residual resistivity of binary alloys to the Büttiker’s voltage-probe model that has been

developed to treat phase breaking scattering in mesoscopic systems. The CPA is typically

applied to treat the conductivity due to elastic scattering originating from substitutional disorder.

For a given configuration, the electronic transport remains coherent. Configurational averaging

replaces the original phase-coherent system by the one involving energy level broadening similar

to that occurring as a result of the coupling to reservoirs that breaks coherence in electron

transmission and produces inelastic scattering. In that sense, the CPA has an analogy to the

Büttiker’s voltage-probe model, though the latter was introduced for a different purpose, namely

to take into account in a simple way phase breaking scattering in mesoscopic conductors that is

essential in experimental conditions.

Within both methods just adding on-site self-energies within the Landauer-Büttiker

approach for conductance would lead to current dissipation. To provide the local current

conservation the chemical potentials of the voltage probes need to be adjusted to guarantee no

current in the floating electrodes. We have shown that within the CPA this procedure is

equivalent to taking into account the vertex corrections, and the spatial dependence of the vertex

exactly follows the local chemical potential. This interpretation allows the determination of the

chemical potential profile across a disordered conductor which is useful for analyzing results of

transport calculations within the CPA. In particular, for layered systems with translational

periodicity in the plane of the layers one can introduce the local resistivity which reflects the

distribution of the resistance across the conductor. The method also allows calculating the

interface resistance between disordered layers. This approach has been illustrated by considering

examples of single-layer, bilayer, and trilayer conductors consisting of different disordered

binary alloys within a tight-binding model. The proposed method may be extended to multiband

spin-dependent systems46 and applied to real geometries of disorder multilayers.47

Page 15

15

Acknowledgements

M.Ye.Zh. thanks the Department of Physics and Astronomy at the University of Nebraska-

Lincoln for hospitality during his stay in spring 2011. E.Y.T. thanks Gerrit Bauer for discussing

the results of this work. This research was supported by the National Science Foundation through

the Nebraska EPSCoR Track II project (Grant No. EPS-1010674) and Nebraska MRSEC (Grant

No. DMR-0820521).

* E-mail: tsymbal@unl.edu

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